Should I apply for the Job? Or Save the friendship?

There is nothing wrong with wanting to better yourself. This job opportunity might not even come up again, so it is entirely up to you to decide if you want this position or not. Just because your friend did not go the extra mile to write that grant, doesn't mean that this job should be given to her due to the fact that she is about to lose her job. If your friend really wanted this job, she should have written that grant. It is understandable that you do not want this job opportunity to come between you and your friend but keep in mind that at the end of the day, she does not sign your paycheck, she does not pay your bills, and she does not keep a roof over your head. If I were you, I apply for that job. If this friend of yours were a true friend, she will be happy for you instead of stopping you to better yourself. Good luck.